We thereafter analyse a time-dependent sudden quench situation, with the XY model coupled to a central spin-1/2
system [1]}, [2]}, [3]}. This was studied first by Zanardi and collaborators [4]}
(see also [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}) in the context of the LE. Here, we study this model in
the context of the NC and the FSC and compare these with the LE.
Interestingly, in the thermodynamic limit, we show that for all times, a very generic formula that
holds in most part of the parameter space of the XY model is \({\mathcal {L}}= e^{-{\mathcal {C}}_N}\) . While for small
times we find that \({\mathcal {L}}\equiv e^{-d\tau ^2}\) , this relation is nonetheless challenged at
large times. In this paper, we further obtain the following results.
a) While \({\mathcal {L}}\)  and the derivative of \({\mathcal {C}}_N\)  are indicators
of quantum phase transitions for all times, \(\tau \)  only indicates such
transitions at large times. b) In the thermodynamic limit, \({\mathcal {L}}\)  and
\({\mathcal {C}}_N\)  show enhanced temporal oscillations when one quenches from the critical line,
while such oscillations are completely absent when the quench is on such a line, and this
is explained by studying the nature of quasi-particle excitations in the vicinity of this line.
Finally, we also make some observations about the triangle inequality for \({\mathcal {C}}_N\) . We show
that this might be violated in certain cases and this indicates that the current study
must be further pursued.
